# Identity Property: Definition & Examples

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• 0:00 What is the Identity Property
• 1:25 Why Are Identities…
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Lesson Transcript
Instructor: Karin Gonzalez

Karin has taught middle and high school Health and has a master's degree in social work.

In this lesson, you will learn the definition of the identity property, discover the four different identity properties and examine examples that clearly illustrate these properties. Following this lesson will be a brief quiz to test your knowledge.

## What Is the Identity Property?

An identity is a number that when added, subtracted, multiplied or divided with any number (let's call this number n), allows n to remain the same. The identity will be either 0 or 1, depending on the operation that we are using. In addition and subtraction, the identity is 0. In multiplication and division, the identity is 1. That means that if 0 is added to or subtracted from n, then n remains the same. Also, if n is multiplied or divided by 1, then n remains the same.

The identity property states that when you use an operation to combine an identity with a number (n), the end result will be n:

n + Additive Identity (0) = n

n - Subtractive Identity (0) = n

n * Multiplicative Identity (1) = n

n/ Divisive Identity (1) = n

How do we know which identity property we're using? That's easy! We can just look at the sign being used in each mathematical expression. The additive identity uses the + sign. The subtractive identity uses the - sign. The multiplicative identity uses the * sign and the divisive identity uses the / sign.

## Why Are the Identities Always 0 and 1?

It was mentioned before that the identity is always 0 for addition and subtraction and 1 for multiplication and division. Why is this, you may ask? Let's look deeper at how the operations actually work!

Addition is the process of adding something to something else. So when you add 7 to 0, you're going to get 7. Subtraction is the process of taking away one amount from another amount. So when you subtract 0 from 7, you're going to get 7.

Multiplication is the process of repeated addition. When you multiply a number with another, you are repeatedly adding a number by the number of times stated by the other number. So, if you have 7 * 8, you are repeatedly adding the number 7 eight times. The identity of multiplication is 1, so if you have 7 * 1, you are repeatedly adding 7 one time. That gives us 7.

Division is the process of separating a number into parts. So if we have 7/1, we are separating the number 7 into one part. That would be 7, right? Yes!

If you were to substitute in any other number for 0 or 1 in the above scenarios, the math wouldn't work, which is why the identities are always 0 or 1.

## Examples of the Identity Property

Let's look at some examples for each of the identity properties of addition, subtraction, multiplication and division, starting with the identity property of addition:

Example 1: 100,000,000 + 0 = 100,000,000

Example 2: -67 + 0 = -67

Example 3: (A + B) + 0 = A + B

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